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examples of nowhere dense sets

Note that ℤℤ\mathbb{Z} is nowhere dense in ℝℝ\mathbb{R} under the usual topology: int⁡ℤ¯=int⁡ℤ=∅intnormal-¯ℤintℤ\operatorname{int}\overline{\mathbb{Z}}=\operatorname{int}\mathbb{Z}=\emptyset. Similarly, 1n⁢ℤ1nℤ\frac{1}{n}\mathbb{Z} is nowhere dense for every n∈ℤnℤn\in\mathbb{Z} with n>0n0n>0.

This result provides an alternative way to prove that ℚℚ\mathbb{Q} is meager in ℝℝ\mathbb{R} under the usual topology, since ℚ=⋃n∈ℤ⁢ and ⁢n>01n⁢ℤℚsubscriptnℤ and n01nℤ\displaystyle\mathbb{Q}=\bigcup_{{n\in\mathbb{Z}\text{ and }n>0}}\textstyle{%
\frac{1}{n}}\mathbb{Z}.